70 Essays in Index Number Theory Laspeyres criticized this formula by showing that the index generally changed even if all prices remained constant (i.e. Pd does not satisfy an identity test to use modern terminology). An even more effective criticism of Pd is that it is not invariant to changes in the units of measurement (whereas Pl is invariant). Laspeyres did not write any further papers on index number theory. He wrote papers on economic history, the history of economic thought and on topical economic issues of his time; see Rinne [1981]. Chapter 5 INDEX NUMBERS* W.E. Diewert Selected Works Laspeyres, E., 1863. Geschichte der Volkswirtschaftlichen Anschauungen der Niederländer und ihrer Literatur zur Zeit der Republik, Leipzig. Laspeyres, E., 1871. "Die Berechnung einer mittleren Waarenpreissteigerung," Jahrbücher für Nationalökonomie und Statistik 16, 296-315. References for Chapter 4 Drobisch, M.W., 1871. "Uber die Berechnung der Veränderungen der Waaren-preise und des Geldswerths," Jahrbucher für Nationalökonomie und Statistik 16, 143-156. Rinne, FL, 1981. "Ernst Louis Etienne Laspeyres 1834-1913," Jahrbucher für Nationalökonomie und Statistik 198, 194-216. The index number problem may be phrased as follows. Suppose we have price data pl = (p\,... ,plN) and quantity data xl = (x\,..., xlN) on N commodities that pertain to economic unit i or that pertain to the same economic unit at time period i for i — 1,2,...,/. The index number problem is to find / numbers Pl and / numbers X1 such that (1) PlXl=pl-xl = Y\N p\x\ fori = l,...,J. z—^n— 1 Pl is the price index for period i (or unit i) and X1 is the corresponding quantity index. Pl is supposed to be representative of all of the prices pln, n — 1,..., N in some sense, while X1 is to be similarly representative of the quantities xln, n — 1,..., N. In what precise sense Pl and X1 represent the individual prices and quantities is not immediately evident and it is this ambiguity which leads to different approaches to index number theory. Note that we require that the product of the price and quantity indexes, PlXl, equals the actual period (or unit) i net expenditures on the N commodities, pl ■ xl. Thus if the Pl are determined, then the X1 may be implicitly determined using equations (1), or vice versa. Each individual consumes the services of thousands of commodities over a year and most producers utilize and/or produce thousands of individual products and services. Index numbers are used to reduce and summarize this overwhelming abundance of microeconomic information. Hence index numbers intrude themselves on virtually every empirical investigation in economics. Index number theory splits naturally into two divisions, depending on the size of /. If I — 2, so that there are data for only two time periods or two economic units, then we are in the realm of bilateral index number theory while if J > 2, then we are in the realm of multilateral indexes. Bilateral approaches are considered in Sections 1-5 below and multilateral approaches are considered in Sections 6-10. *First published in The New Palgrave: A Dictionary of Economics, Vol. 2, J. Eatwell, M. Milgate and P. Newman (eds.), The Macmillan Press, 1987, pp. 767-780. 72 Essays in Index Number Theory 5. Index Numbers 73 The four main approaches to index number theory are: (i) statistical (Section 1), (ii) test or axiomatic (Sections 2 and 9), (iii) microeconomic which relies on the assumption of maximizing or minimizing behavior (Sections 3, 4 and 5), and (iv) neostatistical (Section 10). 1. Statistical Approaches Let 1 — 2 and consider the following formula for P2/P1 due originally to Dutot [1738]: w p'ipl = (t!"„l"'/n) I (t!"„^/n) ■ Thus the average level of prices in say period 2 relative to period 1 is set equal to the arithmetic average of the period 2 prices divided by the arithmetic average of the period 1 prices. The right hand side of (2) is called an index number formula. Given an index number formula, we may solve the aggregation problem (1) as follows: set P2 equal to the index number formula and determine P1, X1 and X2 by: (3) P1 = 1, X1=p1-x1 and X2=p2-x2/P2. Setting P1 — 1 is regarded as an arbitrary normalization; any other convenient normalization such as P1 — 100 could be chosen, in which case X1 = p1 -x1 /P1, p2 = (P2/P1)P1 and X2 = p2 ■ x2/P2, where P2/P1 is the index number formula. Rather than taking P2/P1 to be a ratio of average prices, Carli [1764] suggested taking an average of the price ratios as follows: (4) P'/P1^^ M/P^/N. z—^n— 1 The average of the price ratios in (4) is an arithmetic average. Jevons [1865] suggested using a geometric average: n (5) P'/P^U^/Pn)1^- n=l Once the ratio P2/P1 has been determined by (4) or (5), P1, X1 and X2 may be determined using (3). But which of the three alternative formulae for P2/P1 should we use? Walsh [1901] criticized the use of the Dutot formula (2) on the grounds that the index was not invariant to changes in the units of measurement. This criticism was a telling one, and virtually nobody uses formula (2) at present. However, formulae (4) and (5) are invariant to changes in the units of measurement, so we must still discriminate between them. Jevons argued that changes in the quantity of money between the two periods would lead to proportional changes in all prices except for random errors. In particular, Jevons argued that the price ratios, Pn/Pn, would be independently and symmetrically distributed around a common mean. If this distribution happened to be the normal distribution, then the maximum likelihood estimator for the common mean leads to the index number formula (4). If the ratios p2njp\ happened to be log normally distributed, then statistical considerations would lead us to the index number formula (5). Bowley [1928] attacked the use of both (4) and (5) on two grounds. First, from an empirical point of view, he showed that price ratios were not symmetrically distributed about a common mean and their logarithms also failed to be symmetrically distributed. Secondly, from a theoretical point of view, he argued that it was unlikely that prices or price ratios were independently distributed. Keynes [1930] developed Bowley's second objection in more detail; he argued that changes in the money supply would not affect all prices at the same time. Moreover, real disturbances in the economy could cause one set of prices to differ in a systematic way from other prices, depending on various elasticities of substitution and complementarity. In other words, prices are not randomly distributed, but are systematically related to each other through the general equilibrium of the economy. The above criticisms led to a movement away from the use of unweighted averages of price ratios to represent price movements independently of quantity movements. Walsh [1901] and others suggested that the quantity observations xln that were associated with the individual price observations pln should be used as weights in the price index formula. Scrope [1833] suggested the following formula: (6) P2/P1 =p2 -x/p1 -x where x = (x\,... ,xn) was a somewhat vaguely specified quantity vector which was used to weight the price vectors p1 and p2 as in (6). Laspeyres [1871] recommended that x be set equal to x1, the period 1 quantity vector, while Paasche [1874] suggested that x be set equal to x2, the period 2 quantity vector. This led to the following formulae: (7) P2/Px =p2 -x^p1 -x1 = PL{p\p2,x\x2); (8) P2/P1 = p2 ■ x2/p1 ■ x2 = Ppip^p2^1^2). 74 Essays in Index Number Theory 5. Index Numbers 75 Note that we are now following custom in the index number literature by defining an index number formula to be a function P(p1, p2, x1, x2) of the price and quantity vectors that pertain to the two observations or time periods under consideration: Pp defines the Laspeyres price index while Pp defines the Paasche price index. Pigou [1912] and Irving Fisher [1922] advocated taking a geometric mean of the Paasche and Laspeyres indexes and the resulting formula has come to be known as the Fisher ideal price index Pp: (9) P2/P1 = [PLPp]1/2 = PF{p\p2,x\x2). Rather than taking a geometric average of (7) and (8), Walsh [1901] [1921] advocated using formula (6) where the weight vector x was chosen to be the vector of geometric means of the two quantity vectors: p2ipl - J2N, A&ti)1'2/!?, pic«)1/2 z—' n—1 / z—' n—1 (10) =Pw{p\p2,x\x2). Tornqvist [1936] advocated a weighted geometric mean of the price ratios of the following form: n (11) P'/P1 = II(Pn/pi)s" = Pt(p\p2,x\x2) n=l where sn = (\/2)(p]lx1/p1 -x1) + (\/2)(p2nx2n/p2-x2) is the average expenditure share on good n for n — 1,..., N. It turns out that formulae (7), (8), (9) and (11) are the most widely used formulae for a price index. However, at this point, we have no way of justifying their popularity. Walsh [1901] and Fisher [1922] present scores of functional forms for price indexes — on what basis are we to choose one as being better than the other? This question leads us to discuss the test or axiomatic approach to index number theory. 2. The Test Approach to Bilateral Indexes Consider P(p1,p2,x1, x2), a function of the N period i prices, p1 = (p\,..., plN), and the N period i quantities, xl = (x\,..., xlN) for i — 1,2. The price index P is supposed to represent the level of prices in period 2 relative to period 1. What properties or tests should such an index number formula satisfy? The following nine tests (or closely related variants) have been considered in the literature. BT1: Identity Test: P(p1,p2, ax1, (3x2) — 1 for all numbers a > 0, (3 > 0 if p1 — p2 and x1 — x2. BT2: Proportionality Test: P(p1, ap2, x1, x2) — aP(p1,p2, x1, x2) for a > 0. BT3: Invariance to Changes in Scale Test: P(ap1, ap2, fix1,7a;2) — Pip1 ,p2,x1 ,x2) for all a > 0, j3 > 0 and 7 > 0. BT4: Invariance to Changes in Units (Commensurability) Test: P(aip\,..., aNpxN; aip\,..., aNp2N; a 1 x^, • • • , ax! a^ x^,..., a^ ^jy) — P(p1 ,p2 jX1, x2) for a\ > 0,..., a^v > 0. BT5: Symmetric Treatment of Countries or Time (Country or Time Reversal) Test: P(p2, p1, x2, x1) — 1/P(p1, p2, x1, x2). BT6: Symmetric Treatment of Commodities (Commodity Reversal) Test: P{p1,p2, x1, x2) — P(p1,p2, x1^2) where p% denotes a permutation of the elements of the vector pl and xl denotes the same permutation of the elements of xl, i — 1,2. BT7: Monotonicity Test: P(p1,p2,x1,x2) < P(p1,p3,x1,x2) Hp2 2, this formula, which equals the price ratio for commodity 1 only, fails BT6. BT7 says that if period 2 prices increase in any manner, then the price index cannot decrease. BT8 says that the price index should lie between the smallest and largest price ratios over all commodities. BT9 is a transitivity test which looks beyond the case of only two periods or countries. BT9 says that if we have price and quantity data for three time periods, then the product of the price index going from period 1 to period 2 times the price index going from period 2 to 3 should equal the price index going from period 1 to 3 directly. All of the above tests seem to be reasonable and desirable. If N — 1 so that there is only one commodity, then BT1 and BT2 imply that P{p1,p2, x1, x2) must equal p\jp\ and, of course, this index formula will satisfy all of the remaining tests. In the general N commodity case (assuming that all prices and quantities are positive), which tests are satisfied by the index number formulae defined in the previous section? It can be shown that the Dutot index (2) satisfies all tests except BT4 (but this is a fatal flaw), the Carli index (4) fails only BT5 and BT9, the Jevons geometric index (5) satisfies all tests, the Laspeyres and Paasche indexes defined by (7) and (8) fail BT5 and BT9, the Fisher and Walsh indexes (9) and (10) fail only BT9, and the Tornqvist index fails BT7 and BT9. Thus from the viewpoint of the test approach, it would appear that the geometric index is best. The above conclusion is not warranted since our list of desirable tests is incomplete. We shall consider an additional two tests where the geometric index receives a failing grade. Consider an index number formula that utilizes positive price and quantity information for N commodities, PN(p1 ,p2, x1, x2) say. Now consider the same functional form that uses information on only the first N — 1 commodities, Pw_1 say. Then we may want the index number formula to satisfy the following property: (12) lim PN(p1,p2,x1,x2) — P (Pii ■ ■ ■ jPn — 1>Pi> ■ ■ ■ jPn — 1> xl> ■ ■ ■ ' xn — li xl> ■ ■ ■ ' xn — l)- Thus as the quantity of commodity N tends to 0 in both periods (and thus commodity N becomes irrelevant), the N commodity price index tends to the N — 1 commodity price index, where the prices and quantities of good N have been deleted from the formula. This might be called the irrelevance of tiny commodities test, test BT10. Obviously, this test only makes sense if the basic index number formula P satisfies BT6, so that all commodities are treated symmetrically. The geometric price index (5) fails test BT10 as do the other unweighted formulae (2) and (4). However, the quantity weighted price indexes, (7)—(11), all pass this test. Thus from the viewpoint of passing tests, the Fisher and Walsh indexes, (9) and (10), now look just as good as the geometric index (5). There is another reason for not preferring the geometric price index. Recall our basic aggregation problem (1). It is clear that we can interchange the role of prices and quantities in the two periods and define a quantity index Q(p1,p2,x1,x2) in much the same way that we defined the price index P(p1,p2,x1,x2). We set P2/P1 — P(p1,p2,x1,x2) and we may set X2/X1 — Q(p1,p2,x1,x2). From (1), we deduce that the product of the price and quantity indexes should equal the value ratio for the two periods; i.e. P and Q should satisfy the following product test due to Fisher [1911] but named by Frisch [1930]: (13) Pip1 ,p2, x1, x^Qip1 ,p2, x1, x2) =p2 ■x2/p1 -x1. However, rather than defining Q independently of P, (13) may be rearranged to yield a definition of Q in terms of P. The resulting quantity index is called the implicit quantity index that corresponds to P. If we define the implicit quantity index, Qq say, which corresponds to the geometric price index (5) and consider the quantity counterparts to tests BT1-BT9 above (BT10 does not have a sensible quantity counterpart), we find that Qq fails the quantity counterpart to BT8, the mean value test. Hence we have another reason for failing the geometric price index and its corresponding implicit quantity index. On the other hand, the implicit Fisher quantity index satisfies tests BT1-BT8 adapted for the quantity context. The implicit Fisher quantity index Qp is: QfO?1,?2,^1,^2) = p2 ■ x2/p1 ■ x1 Ppip1 ,p2, x1, x2) — (p1 ■ x2p2 ■ x2/p1 ■ x1p2 ■ x1)1/2 using (9) (14) =PF(x1,3?,p\p2). Thus Qf has the same functional form as Pp except that the role of prices and quantities has been interchanged. We have shown that Pp and Qf satisfy Fisher's [1922] factor reversal test which may be stated as follows: (15) P(p1,p2,x1,x2)P(x1,x2,p1,p2) = p2 ■x2/p1 -x1. 78 Essays in Index Number Theory 5. Index Numbers 79 Of the price indexes defined in the previous section, only the Fisher price index satisfies (15). However, the Fisher price index is by no means the only index number formula that satisfies the factor reversal test. Walsh [1921] showed how to generate hundreds of formulae that would satisfy the test: take a price index P(p1,p2,x1,x2), and define its factor antithesis by p2 ■ x2J p1 ■ x1P(x1,x2,p1,p2). Define a new price index by taking the square root of the product of the original index and its factor antithesis. This new index will automatically satisfy the factor reversal test. The consistency and independence of various bilateral index number tests was studied in some detail by Eichhorn and Voeller [1976]. Our conclusion at this point echoes that of Frisch [1936]: the test approach to index number theory, while extremely useful, does not lead to a single unique index number formula. Thus we turn to economic approaches to index number theory to see if we are led to a more definite conclusion. 3. Microeconomic Approaches to Price Indexes Before a definition of a microeconomic price index is presented, it is necessary to make a few preliminary definitions. Let F(x) be a function of N variables, x = (x\,..., xn). In the consumer context, F represents a consumer's preferences; i.e. if F(x2) > F(xx), then the consumer prefers the commodity vector x2 over x1. In this context, F is called a utility function. In the producer context, F(x) might represent the output that could be produced using the input vector x. In this context, F is called a production function. In order to cover both contexts, we follow the example of Diewert [1976a] and call F an aggregator function. Suppose the consumer or producer faces prices p = {p\,... ,pn) for the N commodities. Then the economic agent will generally find it is useful to minimize the cost of achieving at least a given utility or output level u; we define the cost function or expenditure function C as the solution to this minimization problem: (16) C{u,p) = minimum{p ■ x : F(x) > u} X where p ■ x = ^2^=1 Pnxn is the inner product of the price vector p and quantity vector x. Note that the cost function depends on 1 + N variables; the utility or output level u and the N commodity prices in the vector p. Moreover, the functional form for the aggregator function F completely determines the functional form for C. We say that an aggregator function is neoclassical if F is: (i) continuous, (ii) positive; i.e. F(x) > 0 if x On (which means each component of x is positive), and (iii) linearly homogeneous; i.e. F(Xx) — XF(x) if A > 0. If F is neoclassical, then the corresponding cost function C{u,p) equals u times the unit cost function, c{p) = C(l,p), where c{p) is the minimum cost of producing one unit of utility or output; i.e., (17) C{u,p)=uC{l,p) = uc{p). Shephard [1953] formally defined an aggregator function F to be homothetic if there exists an increasing continuous function of one variable g such that g[F(x)] is neoclassical. However, the concept of homotheticity was well known to Frisch [1936] who termed it expenditure proportionality. If F is homothetic, then its cost function C has the following decomposition: C{u, p) = min{p • x : F{x) > u} X — min{p • x : g[F(x)] > g(u)} X (18) = g{u)c{p) where c{p) is the unit cost function that corresponds to g[F{x)\. Let p1 ^> Oat and p2 On be positive price vectors pertaining to periods or observations 1 and 2. Let x > On be a nonnegative, nonzero reference quantity vector. Then the Koniis [1924] price index or cost of living index is defined as: (19) PK(p\p2,x) = C[F(x),p2]/C[F(x),p1}. In the consumer (producer) context, Pk may be interpreted as follows. Pick a reference utility (output) level u = F(x). Then Pxip1 ,p2 ,x) is the minimum cost of achieving the utility (output) level u when the economic agent faces prices p2 relative to the minimum cost of achieving the same u when the agent faces prices p1. If N — 1 so that there is only one consumer good (or input), then it is easy to show that Pk(p\,p2,xi) — p\x\jp\x\ — p\/p\. Using the fact that a cost function is linearly homogeneous in its price arguments, it can be shown that Pk has the following homogeneity property: Pk (p1, Xp2, x) — XPKip1 ,p2 ,x) for A > 0 which is analogous to the proportionality test BT2 in the previous section. Pk also satisfies Pk(p2,px ,x) — 1/Pk(p1^2, x) which is analogous to the time reversal test, BT5. Note that the functional form for Pk is completely determined by the functional form for the aggregator function F which determines the functional form for the cost function C. In general, Pk depends not only on the two price vectors p1 and p2, but also on the reference vector x. Malmquist [1953], Pollak [1971a] and Samuelson 80 Essays in Index Number Theory 5. Index Numbers 81 and Swamy [1974] have shown that Pk is independent of x and is equal to a ratio of unit cost functions, c(p2)/c(p1), if and only if the aggregator function F is homothetic. If we knew the consumer's preferences or the producer's technology, then we would know F and we could construct the cost function C and the Koniis price index Pk- However, we generally do not know F ov C and thus it is useful to develop bounds that depend on observable price and quantity data but do not depend on the specific functional form for F or C. Samuelson [1947] and Pollak [1971a] established the following bounds on Pk- Let p1 ^ Oat, and p2 On- Then for every reference quantity vector x > Oat, we have (20) mmipl/pl} < PKip1^2, x) < max{j£/p^}; n n i.e., Pk lies between the smallest and largest price ratios. Unfortunately, these bounds are usually too wide to be of much practical use. To obtain closer bounds, we now assume that the observed quantity vec- tors for the two periods, — (x 1; ■ n ), i — 1,2, are solutions to the producer's or consumer's cost minimization problems; i.e., we assume: (21) tf -xl = C(F(xl),tf), ^»Oat, > 0 n, 1,2. Specifically, Koniis showed that there exists a A between 0 and 1 such that if Pp Pl, then (26) Pp < Pk[p\p2, Ax1 + (1 - X)x2] < PL Pl < Pk[p\p2, Ax1 + (1 - X)x2] < PP. The bounds on the microeconomic price index Pk given by (20) and (22)-(26) are the best bounds that we can obtain without making further assumptions on F. In the time series context, the bounds given by (25) or (26) are quite satisfactory: the Paasche and Laspeyres price indexes for consecutive time periods will usually differ by less than 1 percent. However, in the cross section context where the observations represent, for example, production data for two producers in the same industry but in different regions, the bounds are often not very useful since Pl and Pp can differ by 50 percent or more in the cross sectional context; see Ruggles [1967]. In Section 5 below, we will make additional assumptions on the aggregator function F or its cost function dual C that will enable us to determine Pk exactly. Before we do this, in the next section, we will define various quantity indexes that have their origins in microeconomic theory. Given the above assumptions, we now have two natural choices for the reference quantity vector x that occurs in the definition of Pk{pX,p2,x) ■ x1 or x2. The Laspeyres-Koniis price index is defined as Pk^P1 ,p2 ,xx) and the Paasche-Koniis price index is defined as Pk(j>x ,p2 ,x2\ Under the assumption of cost minimizing behavior (21), Koniis [1924] established the following bounds: (22) (23) PK{p\p2^x1) < p2 ■ x1 /p1 ■ x1 = PL{p\p2,x\x2)-PK(p\p2, x2) > p2 ■ x2/p1 ■ x2 = Pp(p\p2, x\x2), where Pl and Pp are the Laspeyres and Paasche price indexes defined earlier by (7) and (8). If in addition, the aggregator function is homothetic, then Frisch [1936] showed that for any reference vector x > Oat, (24) PP =p2 ■x2/p1 -x2 < PKip1^2^) F(x1), then QA(xx,x2,p) > 1 while if F(x2) < F(x1), then QA(xx,x2,p) < 1. Thus the Allen quantity index correctly indicates whether the commodity vector x2 is larger or smaller than x1. It can also be seen that QA satisfies a counterpart to the time reversal test; i.e., QA(x2, xx,p) — l/Q/Jjc1, x2,p). Just as the price index Pk depended on the unobservable aggregator function, so also do the quantity indexes Qk and QA. Thus it is useful to develop bounds for the quantity indexes that do not depend on the particular functional form for F. Samuelson [1947] and Allen [1949] established the following bounds for (28) and (29): (31) QA(xx ,x2 ,px) — QK(p1,p2,x1,x2,x2) < p1 ■ x2Ip1 ■ x1 = QL; (32) QA(x1,a;2,p2) = QK(p\p2, x1, x2, x1) > p2 ■ x2/p2 ■ x1 = QP. Note that the observable Laspeyres and Paasche quantity indexes, Ql and Qp, appear on the right hand sides of (31) and (32). Diewert [1981a], utilizing some results of Pollak [1971a] and Samuelson and Swamy [1974], established the following results: if the underlying aggregator function F is neoclassical and (21) holds, then for all p 3> Oat and x 3> Oat, (33) QP < QA(x\x2,p) = Quip1 ,p2'^x1 ,x2',x) = F(x2)/F(x1) < QL. Thus if the aggregator function F is neoclassical, then the Allen quantity index for all reference vectors p equals the Koniis quantity index for all reference quantity vectors x which in turn equals the ratio of aggregates (F(x2)/F(x1)). Moreover, QA and Qk are bounded from below by the Paasche quantity index Qp, and bounded from above by the Laspeyres quantity index in the neoclassical case. In the general nonhomothetic case, Diewert [1981a] showed that there exists a A between 0 and 1 such that Qk[px,P2, x1, x2, Ax1 + (1 — X)x2] lies between Qp and Ql and there exists a A* between 0 and 1 such that QA[xx, x2, X*px + (1 — X*)p2] also lies between Qp and Ql- Thus the observable Paasche and Laspeyres quantity indexes bound both the Koniis quantity index and the Allen quantity index, provided that we choose appropriate reference vectors between x1 and x2 and p1 and p2 respectively. Using the linear homogeneity property of the cost function in its price arguments, we can show that the Koniis price index has the desirable homogeneity property, Pk(p1, Xp1, x) — A for all A > 0; i.e., if period 2 prices are proportional to period 1 prices, then Pk equals this common proportionality factor. It would be desirable for an analogous homogeneity property to hold for quantity indexes. Unfortunately, it is not in general true that Qk^1, Xx1 ,p*- ,p2, x) — A or that QA{xx, Xx1 ,p) — X. Thus we turn to a third microeconomic approach to defining a quantity index which does have the desirable quantity proportionality property. Let x1 and x2 be the observable quantity vectors in the two situations as usual, let F(x) be an increasing, continuous aggregator function, and let x 0 be a reference quantity vector. Then the Malmquist [1953] quantity index Qm is defined as: (34) Quix1^2^) =D[F{x),x2]/D[F{x),x1] where D(u,xl) = maxfc{/j : F(xl/k) > u, k > 0} is the deflation or distance function which corresponds to F. Thus D[F{x), x2] is the biggest number which will just deflate the quantity vector x2 onto the boundary of the utility (or production) possibilities set {z : F(z) > F(x)} indexed by the reference quantity vector x while D[F{x), x1] is the biggest number which will just deflate the quantity vector x1 onto the set {z : F(z) > F(x)} and Qm is the ratio of these two deflation factors. Qm depends on the unobservable aggregator function F and as usual, we are interested in bounds for Qm- Diewert [1981a] showed that Qm satisfied bounds analogous to (20); i.e., (35) min{x2/x^} < Qm{xx,x2,x) < max{x2/x^}. n n It should be noted that we do not require the assumption of cost minimizing behavior in order to define the Malmquist quantity index or to establish the bounds (35). However, in order to establish the following bounds due to Malmquist [1953] for Qm, we do need the assumption of cost minimizing behavior (21) and we require the reference vector x to be x1 or x2: (36) Qm(xx,x2,xx) < p1 ■ x2/p1 ■ x1 = QL; (37) QM{xx,x2,x2) >p2 -x2/p2 -x1 = QP. 84 Essays in Index Number Theory 5. Index Numbers 85 Diewert [1981a] showed that under the hypothesis of cost minimizing behavior, there exists a A between 0 and 1 such that Qm[x1,x2,Xx1 + (1 — X)x2] lies between Qp and Qp. Thus the Paasche and Laspeyres quantity indexes provide bounds for a Malmquist quantity index for some reference indifference or product surface indexed by a quantity vector which is a A, (1 — A) weighted average of the two observable quantity vectors, x1 and x2. Pollak [1971a] showed that if F is neoclassical, then we can extend the string of equalities in (33) to include the Malmquist quantity index Qm (x1 ,x2 ,x), for any reference quantity vector x. Thus in the case of a linearly homogeneous aggregator function, all three theoretical quantity indexes coincide and this common theoretical index is bounded from below by the Paasche quantity index Qp and bounded from above by the Laspeyres quantity index Qp. In the general case of a nonhomothetic aggregator function, our best theoretical quantity index, the Malmquist index, is also bounded by the Paasche and Laspeyres indexes, provided that we choose a suitable reference quantity vector. We noted in the price index context that the Paasche and Laspeyres price indexes were usually quite close in the time series context. A similar remark also applies to the Paasche and Laspeyres quantity indexes. Thus taking an average of the Paasche and Laspeyres indexes, such as the Fisher price and quantity indexes, will generally approximate underlying microeconomic price and quantity indexes sufficiently accurately for most practical purposes. However, this observation does not apply to the cross sectional context, where the Paasche and Laspeyres indexes can differ widely. In the following section, we offer another microeconomic justification for using the Fisher indexes that also applies in the context of making interregional and cross country comparisons. 5. Exact and Superlative Indexes Assume that the producer or consumer is maximizing a neoclassical aggregator function / subject to a budget constraint during the two periods. Under these conditions, it can be shown that the economic agent is also minimizing cost subject to a utility or output constraint. Moreover, the cost function C that corresponds to / can be written as C[f(x),p] — ,f{x)c{p) where c is the unit cost function (recall (17) above). Suppose a price index P(p1,p2, x1, x2) and a quantity index Q(p1 ,p2, x1, x2) of the type considered in Sections 1 and 2 are given. The quantity index Q is defined to be exact for a neoclassical aggregator function / with unit cost dual c if for every p1 3> Ojv, p2 3> On and xl 3> Ojv which is a solution to the aggregator maximization problem m&xx{f(x) : pl ■ x < pl ■ x1} — ,f{xl) > 0 for i — 1, 2, we have (38) Q(p1,p2,x\x2)^f(x2)/f(x1). Under the same hypothesis, the price index P is exact for / and c if we have (39) Pip^p2^1^2)^^2)/^1). In (38) and (39), the price and quantity vectors are not regarded as being independent. The pl can be independent, but the xl are solutions to the corresponding aggregator maximization problem involving pl, for i — 1,2. Note that if Q is exact for a neoclassical /, then Q can be interpreted as a Koniis, Allen or Malmquist quantity index and the corresponding P defined implicitly by (13) can be interpreted as a Koniis price index. The concept of exactness is due to Koniis and Byushgens [1926]. Below, we shall give some examples of exact index number formulae. Additional examples may be found in Afriat [1972b], Pollak [1971a] and Samuelson and Swamy [1974]. Koniis and Byushgens [1926] showed that Irving Fisher's ideal quantity index Qp defined by (14) and the corresponding price index Pp defined by (9) are exact for the homogeneous quadratic aggregator function / defined by (40) /(xi,... ,xN) = (y y anmxnxm) =(xTAx)1/2 \ z—^n—1 z—J where A = [anm] is a symmetric N x N matrix of constants. Thus under the assumption of maximizing behavior, we can calculate f(x2)/f(x1) — Qp and c(p2)Ic(px) — Pp where / is defined by (40) and c is the unit cost function that corresponds to /. The important thing to note is that / depends on N(N+l)/2 unknown anm parameters but we do not need to know these parameters in order to evaluate f(x2)/f(x1) and c(p2)/c(p1). Diewert [1976a] showed that the Tornqvist price index Pp defined by (11) is exact for the unit cost function c{p) defined by: En „JV „JV an\npn + (1/2) } y amnlnpmlnpn where the parameters an and amn satisfy the following restrictions: En „JV an — 1, y ctmn — 0 for m — 1,..., N and n— 1 —'n—l (42) amn — anm for all to, n. Thus we may calculate c(p2)/c(px) — Pp and f (x2)/ f (x1) — p2 -x2/p1 -xxPp = Qp where c is the unit cost function defined by (41), / is the aggregator function 86 Essays in Index Number Theory 5. Index Numbers 87 which corresponds to this c, and Qt is the implicit Tornqvist quantity index. Note that we do not have to know the parameters an and amn in order to evaluate c(p2)/c(p1) and f(x2)/f(x1). The unit cost function defined by (41) is the translog unit cost function defined by Christensen, Jorgenson and Lau [1971]. Since Pp is exact for this translog functional form, Pt is sometimes called the translog price index. Before we present our final example of an exact index number formula, we need to define a family of quantity indexes Qr that depend on a number, r ^ 0: (43) -(p\p2,a;\:z2) = E n sn(xn/xn) ^ 1/r -r/2 -1/r where sln — PnXln/pl ■ xl is the period i expenditure share for good n. For each r^O, define the corresponding implicit price index by: (44) Pr(p1,p2, X1, X2) — p2 ■ X2/p1 ■ X1Qr(p1 ,p2, X1, X2) A bit of algebra will show that when r — 2, P% — Pf, the Fisher price index defined by (9) and when r — \, P\ — Pw, the Walsh price index defined by (10). Diewert [1976a] showed that Qr and Pr are exact for the quadratic mean of order r aggregator function fr defined by: (45) /r(xi,...,xjv) = V V \ z—^ra—1 z—^n— 1 r/2 r/2 1/r where A = [amn] is a symmetric matrix of constants. Thus the Walsh price index Pw is exact for fi{x) defined by (45) when r — 1. Diewert [1974a] defined a linearly homogeneous function / of N variables to be flexible if it could provide a second order approximation to an arbitrary twice continuously differentiable linearly homogeneous function. It can be shown that / defined by (40), c defined by (41) and (42) and fr defined by (45) for each r ^ 0 are all examples of flexible functional forms. Let the price and quantity indexes P and Q satisfy the product test equality, (13). Then Diewert [1976a] defined P and Q to be superlative indexes if either P is exact for a flexible unit cost function c or Q is exact for a flexible aggregator function /. Thus Pp, Pw, Pt and Pr are all superlative price indexes. At this point, it may seem that we are in the same position that we were at the end of Section 2 where we could not find an index number formula that satisfied all reasonable tests; hence we could not single out any formula as being the best. In the present context, we have a similar problem: how are we to discriminate between Pp, Pw and Pt? Fortunately, it does not matter very much which of these formulae we choose to use in applications; they will all give the same answer to a reasonably high degree of approximation. Diewert [1978b] showed that all known superlative index number formulae approximate each other to the second order when each index is evaluated at an equal price and quantity point. This means the Pp, Pw, Pt and each Pr have the same first and second order partial derivatives with respect to all AN arguments when the derivatives are evaluated at a point where p1 — p2 and x1 — x2. A similar string of equalities also holds for the corresponding implicit quantity indexes defined using the product test (13). Empirically, it has been found that superlative indexes typically approximate each other to something less than 0.2 percent in the time series context and to about 2 percent in the cross section context; see Fisher [1922] and Ruggles [1967]. Diewert [1978b] also showed that the Paasche and Laspeyres indexes approximate the superlative indexes to the first order at an equal price and quantity point. In the time series context, for adjacent periods, the Paasche and Laspeyres price indexes typically differ by less than 0.5 percent; hence these indexes may also provide acceptable approximations to a superlative index. Having considered the case of two observations at great length, we now turn our attention to the I observation case. In the next section, we consider the case of I consecutive time series observations of prices and quantities on the same producer or consumer. We shall consider the general case of I observations separated by space (and possibly by time) in Section 9 below. 6. The Fixed Base Versus the Chain Principle Consider the case of I consecutive observations through time on the prices and quantities of N goods utilized by an economic unit, pl = (p\,... ,plN) and x% — (x\, .. ., xlN), i — 1, . . . , I. Suppose that we have decided on a price index P and a quantity index Q where P and Q satisfy (13). This allows us to compare prices and quantities for any two observations. Based on the test approach, we would probably choose P and Q to be the Fisher indexes, Pp and Qp, since they satisfy the most tests. Since the Fisher indexes also lie between the Paasche and Laspeyres bounds derived in Sections 3 and 4 and moreover are superlative indexes, the choice of the Fisher indexes also fits in well with the microeconomic approach to index number theory. However, if we choose any superlative price index P in order to make bilateral comparisons of prices, then we are faced with a problem when I > 3 88 Essays in Index Number Theory 5. Index Numbers 89 because the circular test BT9 will not be satisfied. Thus choose observation 1 as the base and make all price comparisons relative to period 1, so that the relative price level in period i is P(p1 ,pl, x1, xl). Now choose another observation, say /, as the base so that the price level in period i relative to I is P(pI,pl, x1, xl). In order to make the price level in period 1 equal to unity, divide each P(pI,pl, x1, xl) by Pip1, p1, x1, x1). If the circular test held, then the two price series would coincide; i.e., we would have (46) P(p1,pi,x1,xi) = P(pI,pi,xI,xi)/P(pI,p1,xI,x1), i = l,...,I. However, since the circular test does not hold in general, we are faced with a problem: which period should we choose as the base? There are a number of alternative strategies that could be followed, including: (i) choose the first observation as the base, (ii) take an average over all possible choices of the base period, (iii) abandon the use of a bilateral formula and develop an entirely new multilateral approach, or (iv) use the chain principle, which will be explained below. Alternative (i) seems rather arbitrary but it has simplicity to recommend it. Virtually all official time series indexes are constructed using fixed base Paasche or Laspeyres indexes with the base year being changed every 5 to 15 years. Alternative (ii) seems attractive at first glance since it treats each period in a symmetric fashion. The problem with the method is that economic history has to be rewritten every time a new observation is added to the initial I observations. Alternative (iii) may also seem attractive. For example, in making a price comparison between i and j, we may want to utilize the quantity information for all periods, so that the bilateral index number formula P{pl,pJ ,xl ,xJ) could be replaced by P*(pl ). For example, we could use the Scrope index (6) where x could be taken to be the average quantities over all periods, ^2,\=1 xl /1, or we could use the Tornqvist index (11) where sn could be set equal to the average over all periods for the commodity n expenditure share, X)i=i(l/'I)Pnxn/pl ' x%■ Both of these examples lead to indexes that satisfy the circularity property for the original I observations. However, as was the case with alternative (ii), these new multilateral indexes would have to be recomputed as new time series observations become available. Alternative (iv) is to use the chain principle, originally suggested by Marshall [1887], although the term is due to Fisher [1911]. This principle makes use of the natural order provided by the march of time. One first chooses a bilateral index number formula P. The period 1 price level is set equal to unity and the period 2 price level is set equal to P(p1,p2,x1,x2). The period 3 price level is set equal to P(p1 ,p2, x1, x2)P(p2,p3, x2,x3). The period 4 price level is set equal to the period 3 price level times P(p3,p4, x3,x4), and so on. Thus the period i price level is not obtained by the direct comparison of period i prices with period 1 prices, P(p1 ,pl, x1, xl), but rather as the product of the period by period relative price levels; i.e., by travelling along the links of a chain. The chain principle has one substantial disadvantage: if pl — pJ and xl — xJ and periods i and j are not adjacent, then it is not necessarily the case that the measured price level in period i will coincide with the measured price level in period j. However, the chain principle has a number of advantages: (i) no single period is singled out to play an asymmetric role, (ii) the price levels for I periods are not changed as additional periods are added to the data set, (iii) if a good disappears or a new good is introduced so that N, the number of commodities, changes, then the chain price indexes (and the corresponding implicit quantity indexes) will still be comparable for all periods before and after the change in N and (iv) all superlative indexes will closely approximate each other if the chain principle is used, since changes in prices and quantities tend to be small for adjacent time periods. The above mentioned disadvantage of the chain method is due to the lack of circularity in the bilateral formula P(p1 ,p2, x1, x2). However, experience has shown (e.g., see Fisher [1922]) that deviations from circularity for superlative index number formulae are small in the time series context. In fact, one can show that if the bilateral index P is superlative (or equal to the Paasche or Laspeyres index), then P satisfies the circular test to the first order; i.e., the first order derivatives of P(p1 ,p3, x1 ,x3) and of P(p1,p2,x1, x2)P(p2,p3, x2,x3) with respect to the components of p1, p2, p3, x1, x2 and x3 coincide when evaluated at an equal price and quantity point where p1 — p2 — p3 and x1 — x2 — x3. (This is a new result.) Our conclusion at this point is that alternatives (ii) and (iii) do not look very attractive in the time series context: the use of a fixed base or the chain principle seems preferable. However, in the context of cross section data where there is no natural way of ordering the data sequentially, alternatives (ii) and (iii) become much more attractive. We consider genuine multilateral index number formulae in Sections 9 and 10 below. 7. Aggregation Over Consumers Thus far, our discussion of the microeconomic approach to indexes has been limited to the one consumer or producer case (or the case of two consumers or producers who have identical preferences or technologies). In this section, we relax these restrictions and discuss aggregate consumer or household indexes. In the following section, we discuss aggregate output price and quantity indexes. 90 Essays in Index Number Theory 5. Index Numbers 91 Consider the case of two countries or regions (or two time periods for the same country). We suppose that there are Hi households in country i and the preferences of household h in country i are represented by a continuous utility function, Flh(x, zlh), where x = (xi,...,xn) > Ojv is a nonnegative vector of market goods, p1 = (p\,... ,plN) 3> Oat is the corresponding vector of positive prices that each household in region i faces and zlh is a vector of demographic variables or consumption of public goods by household h in country i. The restricted cost or expenditure function of household h in country i is defined as: (47) Clh(ulh,p\ zlh) = min{pl ■ x : Flh(x, zlh) >/}, X i = 1,2; /i= I,...,Hu where ulh is a utility or welfare level. We assume that xlh > On, the observable consumption vector for household h in the country i, solves (47) with ulh — Flh{xlh, zlh) for i = 1, 2 and h = 1,..., H%. We may use the preferences and observed choices of household h in country i to define a Koniis price index Plh for the level of prices in country 2 relative to country 1: (48) P^ip^p2) =Clh(ulh,p2,zlh)/Clh(ulh,p1,zlh), i = 1,2; /i= Suppose that each expenditure function Clh has a translog functional form; i.e., In Clh{u, p, z) is a quadratic function in the logarithms of its variables, similar to the right hand side of (41) except that now {u,p, z) replaces p. We note that we can approximate arbitrary preferences to the second order using this functional form; i.e., the translog restricted expenditure function is a flexible functional form. Caves, Christensen and Diewert [1982b] establish the following result: if household h in country 1 and household k in country 2 have translog expenditure functions with identical coefficients on the second order terms in commodity prices (this forces some similarity in preferences), then the geometric mean of the two theoretical Koniis price indexes, Plh and P2k, equals the observable Tornqvist or translog price index Pt(px,p2, xlh,x2k); i.e., we have (49) [^V.P^V.P2)]1'2 = Pt{p\p2^\x2% h=l,...,Hi; k = l,...,H2. Rather than deal with the individual household indexes plh defined by (48), one can define an average index, where the average is taken over all households in the country. Thus for a nonnegative weights vector a1 = (a\,..., alH) such that 5^/i=i ah ~ 1> define the country i average price index P% as an a1 weighted geometric mean of the individual indexes: Hi (50) Pl{p\p2,al)^\{Plh{p1,p2)<, * = 1.2. h=l The theoretical index Pl defined in (50) utilizes the price vectors p1 and p2 in both countries but utilizes only the preferences of households in country i. As a final bit of averaging, we take the geometric mean of P1 and P2 to obtain a final theoretical price index that treats each household in each country in a symmetric fashion: (51) P{p\p2,a\a2) ee [P\p\p2,a1)P2(p\p2,a2)]1/2. The natural choices for the household weighting vectors a1 and a2 are: (i) democratic weights and (ii) plutocratic weights. In case (i), each household in each country is given an equal weight; i.e., alh = 1 /Hi, i — 1,2, and h — 1,..., Hi. In case (ii), each household gets a weight that is proportional to its share of consumption in its own country; i.e., alh = p1 ■ xlh jpl ■ xl = slh for i — 1,2 and h — 1,..., Hi, where Xi = J^/jLi x%h 1S country i's aggregate consumption vector. If each household has a translog restricted expenditure function with identical coefficients on the second order terms in commodity prices, then making repeated use of (49), we can deduce that the aggregate price index (51) in case (i) is: (52) Pip1^2,!/^,...,!/^, \/H2,...,\/H2) = fl Y[Pt{j>\p2,x™,x™Y/™* h=l fe=l while in case (ii), the aggregate price index is (53) P(p1,p2,s1,s2) = PT(p1,p2,x1,x2) where sl = (s\,... ,slH) is the country i expenditure share vector for households in country i and x% = 5_^/f=i x%h 1S the aggregate country i consumption vector. Thus if individual household consumption data xlh are available, then the democratic aggregate price index defined by (52) can be evaluated as a geometric mean of individual household translog price indexes. If only aggregate country consumption data xl are available, then the plutocratic aggregate price index defined by (53) can be evaluated as a translog price index (recall (11)), using the aggregate consumption vectors x1 and x2 as quantity weights. 92 Essays in Index Number Theory 5. Index Numbers 93 The terms democratic and plutocratic are due to Prais [1959]. For other approaches to aggregate consumer price indexes, see Prais [1959], Pollak [1981] and Diewert [1983a]. We turn now to the construction of an aggregate quantity index for the two periods. Let the preferences of household h in country i be represented by the continuous, increasing utility function Flh{x) where we have absorbed the vector of demographic variables into the function Flh. For i — 1,2, and h — 1,. .., Hi, define the household h in country i deflation function Dlh(u, x) for x > On and u in the range of F by (54) Dih(u, x) = max{5 : Fih{x/5) > u, 5> 0}. s As in Section 5, define the Malmquist quantity index for x* > On relative to x > On using the preferences of household h in country i by: (55) Qih{x,x*,u)=Dih{u,x*)/Dih{u,x), i = 1, 2; h = 1,..., H%. Let the observed consumption vector of household h in country i be xlh > On and define the corresponding utility level by ulh = Flh(xlh). Define the index of average household consumption in country j relative to i by: (56) = y"3^ Q>k (y"'^ xlh/H, x*k,u^/H; i, j = 1, 2, i± j. To explain the meaning of (56), define the country i average or per capita consumption vector by (57) xl^Yl\xlhlH^ i = 1'2- z—^h—1 Then Q12 is an average of the individual country 2 Malmquist indexes Q2fe(a:1, x2k,u2k) which in turn compares the observed consumption vector of household k in country 2 with the average consumption vector x1 for country 1, using the indifference surface through x2k of household k in country 2 as the reference indifference surface. Similarly, Q21 is an average of the individual country 1 Malmquist indexes Qlk(x2, xlk, ulk). Diewert [1986] showed that under the assumption of expenditure minimizing behavior on the part of consumers in both countries, Q12 has the lower bound p2 ■ x2/p2 ■ x1, a Paasche quantity index in per capita quantities, and [Q21] 1 nas the upper bound p1 ■xP'/p1 -x1, a Laspeyres quantity index in per capita quantities. Diewert also shows that there exists a 0 < A < 1 such that AQ12 + (1 — A)(Q21)-1 lies between these per capita Paasche and Laspeyres quantity indexes. This suggests that we can approximate the theoretical index AQ12 + (1 — A)(Q21)-1 by an average of these Paasche and Laspeyres indexes such as the Fisher index Qp(px ,p2 ,xx ,x2) = (p2 -x2/p2 -x1)1/2^1 -x2/p1 -x1)1/2 where the per capita quantity vectors x% are defined by (57). To summarize this section: we have shown that the translog price index Pp and the Fisher quantity index Qp, which had very satisfactory economic interpretations in the case of one consumer, also have reasonable economic interpretations in the many consumer case. 8. Aggregation Over Producers We assume that there are two regions or countries that are to be compared. The two countries could represent the same country at different time periods. The private production sector in country i uses a vector of primary inputs v1 = (v\, • • •, vlM), where vlm is the amount of input m used in country i. These inputs are different types of labor, capital, land and other natural resources. There are N net outputs that can be produced by the private production sector in each country. These goods are different types of consumer and investment goods, exports and imports. The net output vector for country i is yl = (y\,...,yN) and the corresponding price vector is the positive vector wl = (w\,..., wN) On- We assume wl ■ yl = Y^n=i wnVn > 0 for i = 1, 2. If yln < 0, then good n is utilized as an input in country i; this good could be an imported good or it could be an intermediate input that is produced by the government sector in country i. Note that in contrast to the consumer case, we no longer assume that quantity vectors are nonnegative. The technology set for the private production sector in country i is the set Sl = {(yl,v1)}, a feasible set of net output vectors yl and primary input vectors v1. If knowledge is freely transferable across countries, then Sl — S for i — 1,2 so that there is a common technology set across countries. However, we do not require this assumption in what follows. Define country i's private national product function gl by (58) ^(w, v) = max{w ■ y : (y, v) belongs to S1}, i — 1,2. y The number gl(w,v) is the maximum value of outputs (less the value of imports) that the private production sector of country i can produce, given that each producer in the country faces the price vector w and the aggregate private economy has at its disposal the primary input vector v. If each producer faces the same price vector w and behaves competitively, then we do not have to concern ourselves with individual producer output vectors: all that matters is the aggregate net output vector. The national product function was introduced 94 Essays in Index Number Theory 5. Index Numbers 95 into the economics literature by Samuelson [1953-54]; it is sometimes called a variable or restricted profit function or a net revenue function. The Fisher-Shell [1972a] output price index of country 2 relative to 1 using the country i technology set Sl and primary input vector v1 is defined as (59) 1,2. We assume optimizing behavior on the part of producers in each country so that the observed country i price and net output vectors, wl and yl, satisfy wl ■ yl — gl(wl, vl) > 0 for i — 1, 2. Assume that the private national product function gl{w, v) has a translog functional form for each i (recall (41) except that now In gl{w, v) is a quadratic form in the logarithms of wn and vm) and further assume that the coefficients for the quadratic terms in the logarithms of output prices are the same across the two countries. Then Diewert [1986] showed that the geometric mean of the two theoretical output price indexes defined by (59) is exactly equal to the observable translog price index Pp(w1, w2, y1, y2), where Pt is defined by (11); (60) [P^w^w^P2^1^2)]1/2 = PHw^w2^1^2). Thus the translog price index Pt again turns out to have a strong microeco-nomic justification. The Malmquist quantity index has been applied to the problem of constructing output indexes by Caves, Christensen and Diewert [1982b], and Diewert [1986]. It is first necessary to define the country i output deflation function dl: for a primary input vector v and a net output vector y, define (61) d\y, v) = niin{c> : (y/ö, v) belongs to S\ öl > 0}, s 2=1,2 where Sl is the technology set for country i. Thus dl{y, v) denotes the amount the net output vector y must be deflated so that the deflated output vector and the reference input vector v are just on the frontier of the country i production possibilities set Sl. The Malmquist output index of country 2 relative to country 1 using the country i technology and primary input vector v1 is: (62) f{y\y2)^dl{y2y)/dl{y1y), i = 1,2. Assume the observed country i net output vector yl solves the country i private product maximization problem, maxy{wl ■ y : (y,vl) belongs to S1} for i — 1,2. Then Diewert [1986] established the following bounds for the Malmquist indexes Ql defined by (62): (63) (64) f{y\y2) > w1-y2/w1-y1 = QL{w\w2,y\y2); ?{y\y2) < w2-y2/w2 ■ y1 = QP{w\w2,y\y2) where Qp and Qp are the Laspeyres and Paasche quantity indexes. It is also possible to define a deflation function d that uses a convex combination of the input vectors, At>1 + (1 — A)t>2, and a convex combination of the technology sets for the two countries, AS"1 + (1 — A)^2: (65) d(y, A) = min \d : \y/6, [Xv1 + (1 - X)v2}] belongs to XS1 + (1 - A)^2}. The Malmquist X weighted average output index for country 2 relative to 1 may be defined as: (66) \y\y2) = d{y2,X)ld{y\X). Assuming maximizing behavior, Diewert [1986] shows that there exists a A such that 0 < A < 1 and Qx(y2,y2) lies between the Laspeyres and Paasche quantity indexes Ql and Qp defined in (63) and (64). Thus the Fisher quantity index, Qp = (QlQp)1^2, should provide an adequate approximation to the theoretical output index Qx(yx,y2). Our conclusion is that the translog price index Pp and the Fisher quantity index Qp have reasonably strong justifications in the bilateral aggregate private production context. We now turn our attention to the multilateral case. 9. Multilateral Test Approaches We are finally in a position to study the multilateral index number problem which was set out in the introduction. To review the notation, there are I positive price vectors pl = (p\,... ,plN) and I quantity vectors xl = (x\,..., xlN) with p1 ■ xl > 0 for i — 1,..., /. We wish to find 21 positive numbers Pl (price indexes) and X1 (quantity indexes) such that PlXl — p1 ■ xl for i — 1,..., /. The I data points {pl,xl) will typically be observations on production or consumption units that are separated spatially but yet are still comparable. For the sake of definiteness, we shall refer to the I data points as countries. Each commodity n is supposed to be the same across all countries. This can always be done by a suitable extension of the list of commodities. Our first approach to the construction of a system of multilateral price and quantity indexes is based on the use of a bilateral quantity index Q. In this method, the first step is to pick the 'best' bilateral index number formula: e.g., the Fisher index Qp defined by (14) or the implicit translog quantity index defined by Qt(px,p2, x1, x2) = p2 ■ x2/p1 ■ x1Pp(p1,p2,x1, x2) where Pp is defined by (11). Secondly, pick a numeraire country, say country 1, and then 96 Essays in Index Number Theory 5. Index Nunibers 97 calculate the aggregate quantity for each country i relative to country 1 by evaluating the quantity index Q(p1,pl,x1,x'1). In order to put these relative quantity measures on a symmetric footing, we convert each relative to country 1 quantity measure into a share of world quantity by dividing through by 5^fc=i Q(p1>Pk>x1, xk). For a general numeraire country j, define the share of world quantity for country i, using country j as the numeraire country, by: (67) o\{V,x) = Q{V\V\x\x^l Q{p> ,pk,x\xk), i=l, where p = (p1,... ,|/) is the N x I matrix of price data and x = (x1,..., x1) is the N x I matrix of quantity data. Once the numeraire country j has been chosen and the country i shares erf calculated, we may set X1 = a\ and Pl = p1 ■ xljXl for i — 1,..., J. Thus we have provided a solution to the multilateral index number problem (1). Of course, one is free to renormalize the resulting Pl and X1 if desired; i.e., all X1 can be multiplied by a number provided all Pl are divided by this same number. Kravis [1984] calls this method the star system, since the numeraire country plays a starring role: all countries are compared with it and it alone. We shall assume throughout this section that the index number formula Q satisfies the quantity counterpart to the bilateral price tests BT1 through BT6. (The quantity counterpart to BT1 is Q(ap1, /3p2, x1, x2) — liip1—p2 andx1 — x2, the quantity counterpart to BT2 is Q(p1,p2,x1,ax2) — aQ(p1 ,p2,x1,x2) for a > 0, and so on.) This assumption is not restrictive since our best bilateral formulae, Qp and Qp, both satisfy these tests. Of course, the problem with the star system for making multilateral comparisons is its lack of invariance to the choice of the numeraire or star country. Different choices for the base country will in general give rise to different indexes Pl and X1. This problem can be traced to the lack of circularity of the bilateral formula Q: if Q satisfies the time reversal test BT5 and the circular test BT9 for quantity indexes, then cr\ — crk for all i, j and k; i.e., the shares erf defined by (67) do not depend on the choice of the numeraire country j. However, given that the bilateral formula Q does not satisfy the circularity test (as is the case with Qp and Qt), how can we generate multilateral indexes that treat each country symmetrically? Fisher [1922] recognized that the simplest way of achieving symmetry was to average base specific index numbers over all possible bases. Thus define country i's share of world output Si(p, x) by (68) Sl{p,x) = i=i o\{p,x)jI, 1=1, where the a\ are defined by (67). We can now define country i quantities and prices by (69) X1 Si(p,x), Pl=pl-xl/X\ 2 = 1, Fisher [1922] called this method of constructing multilateral indexes the blend method while Diewert [1986] called it the democratic weights method, since each share of world output using each country as the base is given an equal weight in the formation of the average. Of course, there is no need to use an arithmetic average of the erf as in (68); one can use a geometric average: (70) <71{P,X) n °i & x) i=i vi Using (70), the resulting shares no longer sum to one in general, so country i's share of world output is now defined as: (71) SMS^)/Y.L°«M- - = 1.....' If the Fisher index Qp is used in the definition of the a\, then (72) Si(p,x)/Sj(p,x) = 1 V-f \{Qp{pk,pl,xk,x1)/ H QF{pm,p\xm,x3) and in this case, the multilateral method defined by (70) reduces to a method recommended by Elteto and Koves [1964] and Szulc [1964], the EKS method. Instead of using the Fisher formula in (72), Caves, Christensen and Diewert [1982a] advocated the use of the translog quantity index Qp while Diewert [1986] suggested the use of the implicit translog quantity index Qp, since Qp is well defined even in the case where some quantities xln are negative (whereas Qp is not). We call the indexes generated by (69) and (71) for a general bilateral index Q, generalized EKS indexes. When forming averages of the erf as in (68) or (70), there is no necessity to use equal weights: one can define country j's value share of world output as j3j = p1 ■ X-*/^2Ik=1pk ■ xk (this requires all prices to be measured in units of a common currency) and then we may define a plutocratic share weighted average of the cr\: (73) Si(p,y) = y\ , ßj{P,x)crl(p,x). *—'.7 = 1 Diewert [1986] called this method of constructing multilateral indexes the plutocratic weights method. Another multilateral method that is based on a bilateral index Q may be described as follows. Define (74) {p,x) = y\[ [Q{jP ,p\xj ,xi)~1}~1, i = l,...,I. 98 Essays in Index Number Theory 5. Index Numbers 99 If there is only one commodity so that N — 1 and the bilateral index Q satisfies BT1, BT2 and BT3, then a% = [E^iO^/V')-1]"1 = [ EJ=i xj/x*] _1 = x% J Ej=i x^ which is country i's share of world product. In the general case where N > 1, the 'shares' do not necessarily sum up to unity, so it is necessary to normalize them: (75) Si{p,x) = ai{p,x)^^2k=iak{p,x), i = l,...,I. Diewert [1986] called this the own share method for making multilateral comparisons. The above methods for achieving consistency and symmetry rely on averaging over various bilateral index number comparisons. Fisher [1922] realized that symmetry could be achieved by making comparisons with an average; he called this broadening the base. Thus the basket method (which corresponds to Fisher's [1922] formula 6053 and to method 8 described in Ruggles [1967]) may be described as follows. The price level of country i relative to country j is set equal to p1 ■ (Efc=i xk/I)/p' ' (Efc=i ^V-O- This index number formula is a Scrope index (6), where the reference quantity vector is chosen to be the average market basket, ^2kxk/I. The same result could be achieved if we chose x to be the total market basket, ^2kxk. Now define Qji = pi. xl jpi . x3 |p«. (^2k xk) j-pi ■ (Efc ^fe)] to be the implicit output of country i relative to j. Choose a j as a numeraire country and calculate country i's share of world output as: (76) Si[p,x) = x) — 1- MT1: Multilateral Identity Test: S^aip1,..., aip1, Pix1,..., fiix1) — Pi for i — 1,..., I for all cti > 0, (3i > 0 if p1 — ■ ■ ■ — p1, x1 — ■ ■ ■ — x1 and 1^ = 1. MT2: Proportionality Test: For i — 1,..., I and Aj > 0, Si(p, x1,..., x% 1, Xlx\xl+1,...,xI)/Sj(p,x1,...,xl- 1, XiX1, xl+1,..., x1) — XiSi(p,x)/Sj(p,x) for j = 1,... ,i - l,i + 1,... ,1. MT3: Invariance to Changes in Scale Test: S^aip1,..., aip1, fix1,..., f3x!) — Si(p,x) for all cti > 0, (3 > 0, i — 1,... ,1. MT4: Invariance to Changes in Units Test: Si(a±p\,... ,(XnPn', ■■■ aip{,... ,cxnPn; a^1x\, ... , a^x]^; ... ; a^1x{, ... , a^x]^) — Sl(p,x) for i — 1,... ,1 and ai > 0,..., > 0. MT5: Symmetric Treatment of Countries Test: Let p denote a permutation of the I columns of the N x I matrix p, let x denote the same permutation of the I columns of x, and let S{p,x) denote the same permutation of the I columns of the row vector S{p, x). Then S{p, x) — S{p, x). 100 Essays in Index Number Theory 5. Index Nunibers 101 MT6: Symmetric Treatment of Commodities Test: Let p denote a permutation of the N rows of p and let x denote the same permutation of the N rows of x. Then Si(p, x) — Si(p, x) for i — I,..., I. MT7: Country Partitioning Test: Let Sj = Sj(px,... ,p*; x1,..., x1) for j — 1,. ..,1 and let 0 < A4 < 1. Define Sj+1 = Sjlp1,... ,p1,pl; x1,... ,x%~1, X,x\ xl+1,..., x1, (1 - Xi)xl] for j = 1,...,/,/ + 1. Then 5/ = S-+1 for j = l,...,i-l,i + l,...,I, S-+1 = XiS? and Sj+l = (l-Ai)S/. This property is to hold no matter which country i is partitioned. The plutocratic weights method fails MT2 and MT3; thus the resulting quantity indexes are not invariant to country inflation rates, a very severe defect. The democratic weights method fails the multilateral proportionality test MT2 and the two consistency in aggregation tests, MT7 and MT8. This method is dominated by the generalized EKS method which fails only MT7 and MT8. The basket and GK methods fail MT2 and MT9: both methods fail BT2 when 1 — 2. When 1 — 2, S2/S1 — p2 ■ x2p1 ■ (x1 + x^/p1 ■ x1p2 ■ (x1 + x2) for the basket method, and S^j S\ — p2 ■ x2 jp1 ■ xxPok(p1, p2, x±, x2) for the GK The functions Sj are the share functions for the initial world economy method where the GK bilateral price index is defined as that consists of I countries. The functions are the share functions for a new world economy, where the original country i with price vector pl and quantity vector xl, has been partitioned into two countries with price vectors p1 and p1 and quantity vectors XiX1 and (1 — Xi)xl. MT7 says that under these conditions, the original country i share S( splits into AiSf and (1 — Xi)S( and the remaining shares are unaffected. MT8: Irrelevance of Tiny Countries Test: Let Xi > 0 and define Sj (Xi) = Sj(p,x1, ... , x1^1, \iXl,xl+1, ... ,xT) for j — 1,... ,1. Define Sj~x = Sj(p1, ... , p1^1, pl+1, ■ ■ ■ jP1; x1,..., xl~x, xl+1,..., x1) for j — 1,1. Then limA.^o Sj(Xi) — Sj~x for j — — 1 and ]im.xi—>oSj(Xi) — SjZi for j — z + 1, z + 2, This property is to hold for all choices of the disappearing country i. In the above test, the quantity vector for country i is deflated down to a zero vector. Consider the resulting system of limiting share functions. For all countries except i, the limiting share is equal to the share we would get if we simply deleted the data for country i and defined a system of share functions for a world economy consisting of only I — 1 countries. MT8 is a country counterpart to the irrelevance of tiny commodities test, (12). The above multilateral tests were proposed by Diewert [1986]. Our final multilateral test is a new one. MT9: Bilateral Property Test: When 7 = 2, S2(p1,p2, x1,x2)/S1(p1,p2, xx,x2) satisfies the bilateral tests BT1-BT6 for quantity indexes. When 1 — 2, the multilateral system collapses down to a bilateral system. Hence it seems perfectly sensible to demand that S2 / S1 should satisfy tests BT1-BT6 at least, since our best bilateral indexes, Qf and Qt, satisfied these tests. The seven multilateral methods mentioned above satisfy most of the multilateral tests, assuming that the bilateral index Q satisfies BT1-BT6 (recall that all of the multilateral methods utilize a bilateral index Q except the basket and GK methods). The star system fails MT5 (it is obviously not symmetric) and it fails MT8 when the tiny country is chosen to be the numeraire country. (80) PgjsV.p2,*1,*2)^ / j t PnXnXnl\Xn Xn) / z—' n—1 / E n k=l Pkxkxk/(xk + xk) The five multilateral methods that use a bilateral index Q as a building block all have the property that S2/S1 — Q when 1—2, which explains why these methods pass MT9. Unfortunately, none of the above multilateral methods satisfies all nine multilateral tests. Our tentative conclusion is that if a symmetric multilateral method is desired, then the choice seems to be between the EKS (which fails MT7 and MT8, the consistency in aggregation properties) and the own share method (which fails the multilateral proportionality test, MT2). However, the systematic study of multilateral methods has only begun, so it may well be that better methods will be discovered in the future. We turn now to another class of methods for constructing multilateral indexes. 10. Neostatistical Approaches to Multilateral Indexes In one of the early statistical approaches to the construction of a price index, P2/P1, the index was found by minimizing the sum of squared residuals, Zm=i(Pn/Pn ~ P2/P1)2 with respect to P2IP1. The resulting price index turns out to equal (4). Note that this price index was defined independently of quantities. Theil [1960] initiated a neostatistical approach where the price and quantity indexes, Pl and X1, are simultaneously determined. Theil's best linear price and quantity indexes may be found by solving the following constrained minimization problem: (81) mm P1,...^1 ,x1,...,xJ 102 Essays in Index Number Theory 5. Index Numbers 103 subject to a normalization on the P% or X% such as (82) Yl ! X% = which means that we can interpret the Theil X1 as shares. We may define eij = p1 ■ x3 — PlX3 as an error and then the interpretation of (81) becomes straightforward: we choose the Pl and X1 to minimize the sum of squared errors subject to (82). The Theil index numbers have not been widely used, since they do not satisfy the product test equalities: (83) PlXl=pl-xl, i = l,...,I. Kloek and de Wit [1961] suggested a number of variants of the Theil indexes including one where the constraints (83) were imposed. Thus define the Kloek and de Wit multilateral indexes as the Pl and X1 which solve (81) subject to (82) and (83). Unfortunately, the resulting indexes do not have very satisfactory properties: they fail the multilateral tests MT2, MT3, MT7, MT8 and MT9 (when 1 — 2, the resulting bilateral quantity index fails BT2 and BT3 for quantity indexes.) Another neostatistical approach has been suggested by van Yzeren [1956] which he called the balanced method. The price indexes Pl are determined (up to a normalization or factor of proportionality) by solving the following minimization problem: (84) min The X1 are then determined by (83) and the price normalization may be chosen so that (82) is satisfied. In this case, the errors may be defined by (PlX3)~1(pl ■ x3 jp> ■ x3)P3 X3 — 1 + and the Pl may be found by minimizing Hl=x EJ=1 e« subject to a normalization. The van Yzeren system of share functions does rather well in the multilateral test examination: the share functions pass all tests except the consistency in aggregation tests MT7 and MT8. When 1—2, X2/X1 turns out to equal the Fisher quantity index, Qp. Thus the balanced method satisfies the same tests as the EKS method. However, the EKS method has a more satisfactory economic interpretation and is easier to construct numerically. Many other neostatistical approaches to the construction of multilateral indexes could be explored. However, the resulting methods seem to be rather arbitrary and, moreover, they lack economic interpretations. 11. Other Aspects There are many aspects of index number theory that we cannot cover in this brief survey, such as: (i) sampling problems (see Fisher [1922] and Allen [1975]), (ii) the treatment of seasonality (see Turvey [1979], Balk [1980] and Diewert [1983c]), (iii) consistency in aggregation and the theory of subindexes (see Var-tia [1976a], Pollak [1975] and Diewert [1983a]), (iv) productivity indexes (see Jorgenson and Griliches [1967], Caves, Christensen and Diewert [1982b] and Denny and Fuss [1983]), and (v) econometric approaches to cost of living indexes (see Jorgenson and Slesnick [1983]). However, one area of concern that must be discussed is the new goods problem. Suppose that we are in the time series context and we have price and quantity data for N — 1 commodities in periods 1 and 2, pln and xln for t — 1,2 and n — 1,..., N — 1. Suppose in addition, that x2N units of a new good are sold at the price p2N during period 2. How are we to compute the bilateral price index, P(p1,p2,x1,x2), when we do not know pxN, the price of the new good in period 1? Of course, we can assume xxN — 0, so determining the quantity of the new good in period 1 is no problem. From the viewpoint of the microeconomic approach to index number theory, Hicks [1940] provided a formal solution to this new good problem: if we are in the consumer context, pxN should be the price which would just make the consumer's demand for good N in period 1 equal to zero. The practical problem is that this shadow price is not observable: we require a knowledge of the consumer's indifference surfaces to calculate it. Of course, econometric techniques could be used to estimate these shadow prices (see Diewert [1980] for an example of such a technique in the producer context), but most index number practitioners will find it inconvenient to resort to econometrics. In practice, most official indexes ignore the existence of new goods. In order to illustrate the price index bias that can result from the omission of new goods, we shall present a hypothetical example. Suppose that there are three periods, one 'old' good with constant price and quantity, p\ — x\ — 1 for t — 1, 2, 3 and one 'new' good which appears in period 2, so that x\ — 0. Typically, new goods follow a product cycle: they are introduced at a relatively high price and then the price declines over time. Thus we assume that p\ — 2 and p\ — 1, so that the period 2 price for the new good is twice as high as the period 3 price. We assume that the quantity purchased of the new good in period 2 is / > 0, where / is a fraction which represents the period 2 proportion of new goods to old goods. We assume that the quantity purchased of the new good in period 3 is 2/ and that the shadow price p\ in period 1 that would make the demand for the new good equal to zero is 4. If the new good is ignored, we find that P* — 1 for t — 1, 2, 3 for any reasonable index number formula. The true chain Laspeyres price indexes 104 Essays in Index Number Theory 5. Index Nunibers 105 which do not ignore the new good are: P1 — 1, P2 — 1 and P3 — (1 + /)/(l + 2/). The reader can verify that the same indexes result no matter what shadow price p\ is chosen. Evaluating P3 for various reasonable values of / yields the following period 3 price indexes: if / = 0.01, then P3 = 0.9902; if / = 0.02, then P3 = 0.9808 and if / = 0.05, then P3 = 0.9545. Thus the conventional Laspeyres price index which ignores the existence of new goods will have an upward bias of about 1 to 4.5 percent compared with the true Laspeyres index. In order to evaluate the bias in the conventional chained Paasche and Fisher price indexes, we have to use our assumption that p\ — 4. Under our assumptions, we obtain the following values for the true chained Paasche price indexes in period 3: if / — 0.01, then P3 — 0.9619; if / — 0.02, then P3 = 0.9273 and if / = 0.05, then P3 = 0.8403. We also obtain the following values for the true chained Fisher price indexes in period 3: if f — 0.01, then p3 = 0.9759; if / = 0.02, then P3 = 0.9537 and if / = 0.05, then P3 = 0.8956. Thus the conventional Fisher ideal price index will have an upward bias of about 2.5 to 10.5 percent in period 3, depending on the fraction / of new goods introduced in period 2. The above analysis of bias is only illustrative but it does indicate that ignoring new goods could lead to a substantial overestimation of price inflation and a corresponding underestimation of real growth rates, especially in advanced market economies where millions of new goods are introduced each year. References for Chapter 5 Afriat, S.N., 1972b. "The Theory of International Comparisons of Real Income and Prices." 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